HYACINTHOS 25759

[Antreas P. Hatzipolakis]:

 Let ABC be a triangle and A'B'C' the pedal triangle of N.

    Denote:

    Ab, Ac = the orthogonal projections of A' on NB', NC', resp.
    Mab, Mac = the midpoints of BAb, CAc, resp.
    Ma = the midpoint of MabMac.
    Similarly Mb, Mc

    The centroid of MaMbMc lies on the Euler line.

 

[Angel Montesdeoca]:

   
    ***  The centroid of MaMbMc is W= X(2) + X(10128)
   
    W = ( 2 a^6+a^4 (b^2+c^2)-2 a^2 (b^4-16 b^2 c^2+c^4) -(b^2-c^2)^2 (b^2+c^2): ... : ...),
   
        On the lines  {2,3}, {1503,10219}, {3564,6688}

with (6-9-13)-search numbers  (1.95451272868902, 1.07983675186231, 1.99100239430016).
   
    Angel Montesdeoca